Showing a function is square integrable

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I'm trying to show that the dirac delta function is in $H^frac-n2- epsilon(mathbbR^n) forall epsilon > 0.$ Where $H^s(mathbbR^n)$ denotes Sobolev space of order $s$ on $mathbbR^n$

I have that since $hatdelta = 1 Rightarrow <xi>^shatdelta = <xi>^s$ trivially.

So this amounts to needing to show that $int_mathbbR^n |(1+|xi|^2)^fracs2|^2d xi in L^2(mathbbR^n, s= -fracn2 - epsilon.$

But now I'm stuck. I know I need to show that this integral is finite.

Any help appreciated.

Thanks.

real-analysis functional-analysis proof-explanation sobolev-spaces lp-spaces

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edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

asked Jul 15 at 17:47

VBACODER

748

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  Polar coordinates...
  – David C. Ullrich
  Jul 15 at 17:51
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
  – VBACODER
  Jul 15 at 19:17
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
  – PhoemueX
  Jul 15 at 20:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
  – David C. Ullrich
  Jul 15 at 22:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
  – VBACODER
  Jul 16 at 19:57
  
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

favorite

I'm trying to show that the dirac delta function is in $H^frac-n2- epsilon(mathbbR^n) forall epsilon > 0.$ Where $H^s(mathbbR^n)$ denotes Sobolev space of order $s$ on $mathbbR^n$

I have that since $hatdelta = 1 Rightarrow <xi>^shatdelta = <xi>^s$ trivially.

So this amounts to needing to show that $int_mathbbR^n |(1+|xi|^2)^fracs2|^2d xi in L^2(mathbbR^n, s= -fracn2 - epsilon.$

But now I'm stuck. I know I need to show that this integral is finite.

Any help appreciated.

Thanks.

real-analysis functional-analysis proof-explanation sobolev-spaces lp-spaces

share|cite|improve this question

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

asked Jul 15 at 17:47

VBACODER

748

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  Polar coordinates...
  – David C. Ullrich
  Jul 15 at 17:51
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
  – VBACODER
  Jul 15 at 19:17
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
  – PhoemueX
  Jul 15 at 20:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
  – David C. Ullrich
  Jul 15 at 22:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
  – VBACODER
  Jul 16 at 19:57
  
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

favorite

up vote
1
down vote

favorite

I'm trying to show that the dirac delta function is in $H^frac-n2- epsilon(mathbbR^n) forall epsilon > 0.$ Where $H^s(mathbbR^n)$ denotes Sobolev space of order $s$ on $mathbbR^n$

I have that since $hatdelta = 1 Rightarrow <xi>^shatdelta = <xi>^s$ trivially.

So this amounts to needing to show that $int_mathbbR^n |(1+|xi|^2)^fracs2|^2d xi in L^2(mathbbR^n, s= -fracn2 - epsilon.$

But now I'm stuck. I know I need to show that this integral is finite.

Any help appreciated.

Thanks.

real-analysis functional-analysis proof-explanation sobolev-spaces lp-spaces

share|cite|improve this question

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

asked Jul 15 at 17:47

VBACODER

748

I'm trying to show that the dirac delta function is in $H^frac-n2- epsilon(mathbbR^n) forall epsilon > 0.$ Where $H^s(mathbbR^n)$ denotes Sobolev space of order $s$ on $mathbbR^n$

I have that since $hatdelta = 1 Rightarrow <xi>^shatdelta = <xi>^s$ trivially.

So this amounts to needing to show that $int_mathbbR^n |(1+|xi|^2)^fracs2|^2d xi in L^2(mathbbR^n, s= -fracn2 - epsilon.$

But now I'm stuck. I know I need to show that this integral is finite.

Any help appreciated.

Thanks.

real-analysis functional-analysis proof-explanation sobolev-spaces lp-spaces

share|cite|improve this question

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

asked Jul 15 at 17:47

VBACODER

748

share|cite|improve this question

share|cite|improve this question

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

edited Jul 15 at 17:53

David C. Ullrich

54.3k33583

54.3k33583

asked Jul 15 at 17:47

VBACODER

748

asked Jul 15 at 17:47

VBACODER

748

asked Jul 15 at 17:47

VBACODER

748

748

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  Polar coordinates...
  – David C. Ullrich
  Jul 15 at 17:51
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
  – VBACODER
  Jul 15 at 19:17
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
  – PhoemueX
  Jul 15 at 20:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
  – David C. Ullrich
  Jul 15 at 22:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
  – VBACODER
  Jul 16 at 19:57
  
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  Polar coordinates...
  – David C. Ullrich
  Jul 15 at 17:51
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
  – VBACODER
  Jul 15 at 19:17
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
  – PhoemueX
  Jul 15 at 20:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
  – David C. Ullrich
  Jul 15 at 22:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
  – VBACODER
  Jul 16 at 19:57
  
  

  
  
  
  

5

5

Polar coordinates...
– David C. Ullrich
Jul 15 at 17:51

Polar coordinates...
– David C. Ullrich
Jul 15 at 17:51

@DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
– VBACODER
Jul 15 at 19:17

@DavidC.Ullrich I had an attempt not sure if it's right. Rewriting the integral as $ int_mathbbR^n|(1+|xi|^2)^fracs2|^2 = int_mathbbR^n|(r^2)^fracs2|^2 = int_mathbbR^nr^2sdr $ but then I'm not sure how to evaluate this (if this is even correct).
– VBACODER
Jul 15 at 19:17

@VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
– PhoemueX
Jul 15 at 20:47

@VBACODER: The integral with respect to $r$ should be on $(0,infty)$, not $BbbR^d$.
– PhoemueX
Jul 15 at 20:47

Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
– David C. Ullrich
Jul 15 at 22:35

Find a book on real analysis. If $f(x)=g(|x|)$ then $int_Bbb R^nf(x),dx=c_nint_0^infty g(r) r^n-1,dr$.
– David C. Ullrich
Jul 15 at 22:35

@DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
– VBACODER
Jul 16 at 19:57

@DavidC.Ullrich any recommendations for books that will have this in? I have the Folland book. Using the result in your comment, how would one proceed to get the result? If i integrate by parts i obtain $g(r) fracr^nn|_0^infty$ $ - frac1n int_0^infty r^ng'(r)dr$
– VBACODER
Jul 16 at 19:57

 | 
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